On Measurable Functions Satisfying Multiplicative Type Functional Equations Almost Everywhere

2012 ◽  
pp. 241-253 ◽  
Author(s):  
Antal Járai ◽  
Károly Lajkó ◽  
Fruzsina Mészáros
Keyword(s):  
Functional Equations ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Multiplicative Type

Some Pointwise Convergence Results in Lp(μ), 1 < p < ∞

1977 ◽  
Vol 20 (3) ◽  
pp. 277-284 ◽  
Author(s):  
Richard Duncan
Keyword(s):  
Probability Theory ◽  
Pointwise Convergence ◽  
Function Spaces ◽  
Almost Everywhere Convergence ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Convergence Results ◽  
Convergence In Norm

The theory of almost everywhere convergence has its roots in the poineering work of A. Kolmogorov, and today it constitutes one of the most captivating and challenging chapters in modern probability theory and analysis. Whereas some modes of convergence for sequences of measurable functions, e.g. convergence in norm, can be readily obtained by an intelligent exploitation of the various properties of the function spaces involved, a.e. convergence invariably requires a rather high, and sometimes surprising, degree of mathematical virtuosity.

scholarly journals On functional equations of the multiplicative type

2015 ◽  
Vol 431 (1) ◽  
pp. 283-299
Author(s):  
Esteban A. Chávez ◽  
Prasanna K. Sahoo
Keyword(s):  
Functional Equations ◽  
Multiplicative Type

Multiplicative type functional equations arising from characterization problems

2012 ◽  
Vol 83 (3) ◽  
pp. 199-208 ◽  
Author(s):  
Károly Lajkó ◽  
Fruzsina Mészáros
Keyword(s):  
Functional Equations ◽  
Multiplicative Type

scholarly journals The algebras of bounded and essentially bounded Lebesgue measurable functions

2017 ◽  
Vol 50 (1) ◽  
pp. 94-99
Author(s):  
Raymond Mortini ◽  
Rudolf Rupp
Keyword(s):  
Positive Lebesgue Measure ◽  
Discrete Topology ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Bounded Complex ◽  
Elementary Methods ◽  
Point Evaluations ◽  
Complex Valued ◽  
Totally Disconnected ◽  
The Ideal

Abstract Let X be a set in ℝn with positive Lebesgue measure. It is well known that the spectrum of the algebra L∞(X) of (equivalence classes) of essentially bounded, complex-valued, measurable functions on X is an extremely disconnected compact Hausdorff space.We show, by elementary methods, that the spectrum M of the algebra ℒb(X, ℂ) of all bounded measurable functions on X is not extremely disconnected, though totally disconnected. Let ∆ = { δx : x ∈ X} be the set of point evaluations and let g be the Gelfand topology on M. Then (∆, g) is homeomorphic to (X, Τdis),where Tdis is the discrete topology. Moreover, ∆ is a dense subset of the spectrum M of ℒb(X, ℂ). Finally, the hull h(I), (which is homeomorphic to M(L∞(X))), of the ideal of all functions in ℒb(X, ℂ) vanishing almost everywhere on X is a nowhere dense and extremely disconnected subset of the Corona M \ ∆ of ℒb(X, ℂ).

Addition Theorems and Binary Expansions

1995 ◽  
Vol 47 (2) ◽  
pp. 262-273 ◽  
Author(s):  
Jonathan M. Borwein ◽  
Roland Girgensohn
Keyword(s):  
Unknown Function ◽  
Functional Equations ◽  
Elliptic Integral ◽  
Addition Theorems ◽  
Trigonometric Functions ◽  
Binary Expansion ◽  
Almost Everywhere ◽  
Image Position ◽  
Increasing Functions ◽  
Inverse Trigonometric Functions

AbstractLet an interval I ⊂ ℝ and subsets D0, D1 ⊂ I with D0 ∪ D1 = I and D0 ∩ D1 = Ø be given, as well as functions r0: D0 → I, r1: D1 → I. We investigate the system (S) of two functional equations for an unknown function f: I → [0, 1]: We derive conditions for the existence, continuity and monotonicity of a solution. It turns out that the binary expansion of a solution can be computed in a simple recursive way. This recursion is algebraic for, e.g., inverse trigonometric functions, but also for the elliptic integral of the first kind. Moreover, we use (S) to construct two kinds of peculiar functions: surjective functions whose intervals of constancy are residual in I, and strictly increasing functions whose derivative is 0 almost everywhere.

Localising sets for sigma-algebras and related point transformations

1991 ◽  
Vol 118 (1-2) ◽  
pp. 111-118 ◽  
Author(s):  
Alan Lambert
Keyword(s):  
Measurable Function ◽  
Conditional Expectation ◽  
Measure Space ◽  
Functional Equations ◽  
Almost Everywhere ◽  
Weighted Point ◽  
Point Transformations ◽  
Sigma Algebras ◽  
Related Point ◽  
Complete Characterisation

SynopsisEach sigma-finite subalgebra from the sigma-algebra of a measure space induces a conditional expectation operator which acts on L2 as well as the set of almost everywhere nonnegative measurable functions. The concept of localising set is introduced and shown to be closely related to certain functional equations involving . Localising sets are shown to arise naturally in the study of weighted point transformations f→ϕ. f°T, where ϕ is a measurable function and T is a measurable self-map of the state space. A complete characterisation of localising sets related to such transformations is given when the underlying measure space is completely atomic.

scholarly journals Spaces of measurable functions

2013 ◽  
Vol 11 (7) ◽  
Author(s):  
Piotr Niemiec
Keyword(s):  
Hilbert Space ◽  
Measure Space ◽  
Finite Measure ◽  
Metrizable Space ◽  
Equivalence Classes ◽  
Convergence In Measure ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Completely Metrizable

AbstractFor a metrizable space X and a finite measure space (Ω, $\mathfrak{M}$, µ), the space M µ(X) of all equivalence classes (under the relation of equality almost everywhere mod µ) of $\mathfrak{M}$-measurable functions from Ω to X, whose images are separable, equipped with the topology of convergence in measure, and some of its subspaces are studied. In particular, it is shown that M µ(X) is homeomorphic to a Hilbert space provided µ is (nonzero) nonatomic and X is completely metrizable and has more than one point.

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